Abstract
We explore an intermediate “Semi-Lagrangian” formulation of Geometric Optics type problems which stands between the classical “Lagrangian” Hamiltonian systems and its “Eulerian” Hamilton–Jacobi Partial Differential Equations counterpart. The goal is to design a numerical method which “trades” between advantages and drawbacks of both worlds to compute numerically the so-called “multi-valued” solution. We present a numerical algorithm and apply it to the “paraxial” simplification of 2-D geometric optics. We discuss its limitations and its possible extensions to higher dimensions. For an extended discussion on Lagrangian versus Eulerian methods in Geometric Optics and a review of some applications of these techniques see Engquist and Runborg, Acta Numer. (2003), and Benamou, J. Sci. Comput., 19 (2003).
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References
V. I. Arnol’d, Catastrophe theory,Springer, Berlin–Heidelberg, 1992.
V. I. Arnol’d, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differential Maps, Birkhäuser, Boston, 1986.
J.-D. Benamou, An introduction to Eulerian geometric optics (1992–2002), J. Sci. Comput., 19 (2003), pp. 63–94.
J.-D. Benamou, Direct solution of multi-valued phase-space solutions for Hamilton–Jacobi equations, Commun. Pure Appl. Math., 52 (1999), pp. 1443–1475.
J.-D. Benamou, O. Lafitte, R. Sentis, and I. Solliec, A geometric optics method for high frequency electromagnetic fields computations near fold caustics – Part I, J. Comput. Appl. Math., 156 (2003), pp. 93–125.
J.-D. Benamou, O. Lafitte, R. Sentis, and I. Solliec, A geometric optics method for high frequency electromagnetic fields computations near fold caustics – Part II, J. Comput. Appl. Math., 167 (2004), pp. 91–134.
J. J. Duistermaat, Oscillatory integrals, lagrange immersions and unfolding of singularities, Commun. Pure Appl. Math., 27 (1974), pp. 207–281.
B. Engquist and O. Runborg, Computational high frequency wave propagation, Acta Numer., 12 (2003), pp. 181–266.
I. M. Gelfand and S. V. Fomin, Calculus of Variation, Prentice-Hall, Englewood Cliffs, NJ, 1963.
S. Izumiya, The theory of Legendrian unfoldings and first order differential equations, Proc. R. Soc. Edinb., Sect. A, Math., 123 (1993), pp. 517–532.
T. Poston and I. N. Stewart, Taylor Expansions and Catastrophes, Pitman, London–San-Francisco–Melbourne, 1976.
L. C. Young, Lecture on the Calculus of Variation and Optimal Control Theory, W. B. Saunders Co., Philadelphia, PA–London–Toronto, ON 1969.
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Dedicated to Björn Engquist on the occasion of his 60th birthday.
AMS subject classification (2000)
49L25, 70H05, 70H20, 78A05
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Benamou, JD. A semi-lagrangian numerical method for geometric optics type problems . Bit Numer Math 46 (Suppl 1), 5–17 (2006). https://doi.org/10.1007/s10543-006-0081-0
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DOI: https://doi.org/10.1007/s10543-006-0081-0